A note on odd partition numbers

Ramanujan’s partition congruences modulo \(\ell \in \{5, 7, 11\}\) $\ell \in \{5,7,11\}$ assert that \(\begin{aligned} p(\ell n+\delta _{\ell })\equiv 0\pmod {\ell }, \end{aligned}\) $$\begin{array}{c}\hfill p(\ell n+{\delta }_{\ell })\equiv 0(mod\ell ),\end{array}$$ where \(0<\delta _{\ell }<\ell \) $0<{\delta }_{\ell }<\ell$ satisfies \(24\delta _{\ell }\equiv 1\pmod {\ell }.\) $24{\delta }_{\ell }\equiv 1(mod\ell ).$ By proving Subbarao’s conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes \(\ell \ge 5,\) $\ell \ge 5,$ we give a new proof of the conclusion that there are infinitely many m for which \(p(\ell m+\delta _{\ell })\) $p(\ell m+{\delta }_{\ell })$ is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such m satisfies \(m<(\ell ^2-1)/24,\) $m<({\ell }^2-1)/24,$ representing a significant improvement to the previous bound.